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Coupled fixed point theorems on partially ordered Gmetric spaces
Fixed Point Theory and Applications volume 2012, Article number: 174 (2012)
Abstract
The purpose of this paper is to extend some recent coupled fixed point theorems in the context of partially ordered Gmetric spaces in a virtually different and more natural way.
MSC:46N40, 47H10, 54H25, 46T99.
1 Introduction and preliminaries
The notion of metric space was introduced by Fréchet [1] in 1906. In almost all fields of quantitative sciences which require the use of analysis, metric spaces play a major role. Internet search engines, image classification, protein classification (see, e.g., [2]) can be listed as examples in which metric spaces have been extensively used to solve major problems. It is conceivable that metric spaces will be needed to explore new problems that will arise in quantitative sciences in the future. Therefore, it is necessary to consider various generalizations of metrics and metric spaces to broaden the scope of applied sciences. In this respect, cone metric spaces, fuzzy metric spaces, partial metric spaces, quasimetric spaces and bmetric spaces can be given as the main examples. Applications of these different approaches to metrics and metric spaces make it evident that fixed point theorems are important not only for the branches of mainstream mathematics, but also for many divisions of applied sciences.
Inspired by this motivation Mustafa and Sims [3] introduced the notion of a Gmetric space in 2004 (see also [4–7]). In their introductory paper, the authors investigated versions of the celebrated theorems of the fixed point theory such as the Banach contraction mapping principle [8] from the point of view of Gmetrics. Another fundamental aspect in the theory of existence and uniqueness of fixed points was considered by Ran and Reurings [9] in partially ordered metric spaces. After Ran and Reurings’ pioneering work, several authors have focused on the fixed points in ordered metric spaces and have used the obtained results to discuss the existence and uniqueness of solutions of differential equations, more precisely, of boundary value problems (see, e.g., [10–20]). Upon the introduction of the notion of coupled fixed points by Guo and Laksmikantham [14], GnanaBhaskar and Lakshmikantham [15] obtained interesting results related to differential equations with periodic boundary conditions by developing the mixed monotone property in the context of partially ordered metric spaces. As a continuation of this trend, many authors conducted research on the coupled fixed point theory and many results in this direction were published (see, for example, [21–35]).
In this paper, we prove the theorem that amalgamates these three seminal approaches in the study of fixed point theory, the so called Gmetrics, coupled fixed points and partially ordered spaces.
We shall start with some necessary definitions and a detailed overview of the fundamental results developed in the remarkable works mentioned above. Throughout this paper, ℕ and ${\mathbb{N}}^{\ast}$ denote the set of nonnegative integers and the set of positive integers respectively.
Definition 1 (See [3])
Let X be a nonempty set, $G:X\times X\times X\to {\mathbb{R}}^{+}$ be a function satisfying the following properties:
(G1) $G(x,y,z)=0$ if $x=y=z$,
(G2) $G(x,x,y)>0$ for all $x,y\in X$ with $x\ne y$,
(G3) $G(x,x,y)\le G(x,y,z)$ for all $x,y,z\in X$ with $y\ne z$,
(G4) $G(x,y,z)=G(x,z,y)=G(y,z,x)=\cdots $ (symmetry in all three variables),
(G5) $G(x,y,z)\le G(x,a,a)+G(a,y,z)$ for all $x,y,z,a\in X$ (rectangle inequality).
Then the function G is called a generalized metric or, more specially, a Gmetric on X, and the pair $(X,G)$ is called a Gmetric space.
It can be easily verified that every Gmetric on X induces a metric ${d}_{G}$ on X given by
Trivial examples of Gmetric are as follows.
Example 2 Let $(X,d)$ be a metric space. The function $G:X\times X\times X\to [0,+\mathrm{\infty})$, defined by
or
for all $x,y,z\in X$, is a Gmetric on X.
The concepts of convergence, continuity, completeness and Cauchy sequence have also been defined in [3].
Definition 3 (See [3])
Let $(X,G)$ be a Gmetric space, and let $\{{x}_{n}\}$ be a sequence of points of X. We say that $\{{x}_{n}\}$ is Gconvergent to $x\in X$ if ${lim}_{n,m\to +\mathrm{\infty}}G(x,{x}_{n},{x}_{m})=0$, that is, if for any $\epsilon >0$, there exists $N\in \mathbb{N}$ such that $G(x,{x}_{n},{x}_{m})<\epsilon $ for all $n,m\ge N$. We call x the limit of the sequence and write ${x}_{n}\to x$ or ${lim}_{n\to +\mathrm{\infty}}{x}_{n}=x$.
Proposition 4 (See [3])
Let $(X,G)$ be a Gmetric space. The following statements are equivalent:

(1)
$\{{x}_{n}\}$ is Gconvergent to x,

(2)
$G({x}_{n},{x}_{n},x)\to 0$ as $n\to +\mathrm{\infty}$,

(3)
$G({x}_{n},x,x)\to 0$ as $n\to +\mathrm{\infty}$,

(4)
$G({x}_{n},{x}_{m},x)\to 0$ as $n,m\to +\mathrm{\infty}$.
Definition 5 (See [3])
Let $(X,G)$ be a Gmetric space. A sequence $\{{x}_{n}\}$ is called GCauchy sequence if for any $\epsilon >0$, there is $N\in \mathbb{N}$ such that $G({x}_{n},{x}_{m},{x}_{l})<\epsilon $ for all $m,n,l\ge N$, that is, $G({x}_{n},{x}_{m},{x}_{l})\to 0$ as $n,m,l\to +\mathrm{\infty}$.
Proposition 6 (See [3])
Let $(X,G)$ be a Gmetric space. The following statements are equivalent:

(1)
The sequence $\{{x}_{n}\}$ is GCauchy.

(2)
For any $\epsilon >0$, there exists $N\in \mathbb{N}$ such that $G({x}_{n},{x}_{m},{x}_{m})<\epsilon $, for all $m,n\ge N$.
Definition 7 (See [3])
A Gmetric space $(X,G)$ is called Gcomplete if every GCauchy sequence is Gconvergent in $(X,G)$.
Definition 8 Let $(X,G)$ be a Gmetric space. A mapping $F:X\times X\times X\to X$ is said to be continuous if for any three Gconvergent sequences $\{{x}_{n}\}$, $\{{y}_{n}\}$ and $\{{z}_{n}\}$ converging to x, y and z respectively, $\{F({x}_{n},{y}_{n},{z}_{n})\}$ is Gconvergent to $F(x,y,z)$.
We define below gordered complete Gmetric spaces.
Definition 9 Let $(X,\u2aaf)$ be a partially ordered set, $(X,G)$ be a Gmetric space and $g:X\to X$ be a mapping. A partially ordered Gmetric space, $(X,G,\u2aaf)$, is called gordered complete if for each Gconvergent sequence ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}\subset X$, the following conditions hold:
($O{C}_{1}$) If $\{{x}_{n}\}$ is a nonincreasing sequence in X such that ${x}_{n}\to {x}^{\ast}$, then $g{x}^{\ast}\u2aafg{x}_{n}$ $\mathrm{\forall}n\in \mathbb{N}$.
($O{C}_{2}$) If $\{{x}_{n}\}$ is a nondecreasing sequence in X such that ${x}_{n}\to {x}^{\ast}$, then $g{x}^{\ast}\u2ab0g{x}_{n}$ $\mathrm{\forall}n\in \mathbb{N}$.
In particular, if g is the identity mapping in ($O{C}_{1}$) and ($O{C}_{2}$), the partially ordered Gmetric space, $(X,G,\u2aaf)$, is called ordered complete.
We next recall some basic notions from the coupled fixed point theory. In 1987 Guo and Lakshmikantham [14] defined the concept of a coupled fixed point. In 2006, in order to prove the existence and uniqueness of the coupled fixed point of an operator $F:X\times X\to X$ on a partially ordered metric space, GnanaBhaskar and Lakshmikantham [15] reconsidered the notion of a coupled fixed point via the mixed monotone property.
Definition 10 ([15])
Let $(X,\u2aaf)$ be a partially ordered set and $F:X\times X\to X$. The mapping F is said to have the mixed monotone property if $F(x,y)$ is monotone nondecreasing in x and is monotone nonincreasing in y, that is, for any $x,y\in X$,
and
Definition 11 ([15])
An element $(x,y)\in X\times X$ is called a coupled fixed point of the mapping $F:X\times X\to X$ if
The results in [15] were extended by Lakshmikantham and Ćirić in [16] by defining the mixed gmonotone property.
Definition 12 Let $(X,\u2aaf)$ be a partially ordered set, $F:X\times X\to X$ and $g:X\to X$. The function F is said to have mixed gmonotone property if $F(x,y)$ is monotone gnondecreasing in x and is monotone gnonincreasing in y, that is, for any $x,y\in X$,
and
It is clear that Definition 12 reduces to Definition 10 when g is the identity mapping.
Definition 13 An element $(x,y)\in X\times X$ is called a coupled coincidence point of the mappings $F:X\times X\to X$ and $g:X\to X$ if
and a common coupled fixed point of F and g if
Definition 14 The mappings $F:X\times X\to X$ and $g:X\to X$ are said to commute if
Throughout the rest of the paper, we shall use the notation gx instead of $g(x)$, where $g:X\to X$ and $x\in X$, for brevity. In [35], Nashine proved the following theorems.
Theorem 15 Let $(X,G,\u2aaf)$ be a partially ordered Gmetric space. Let $F:X\times X\to X$ and $g:X\to X$ be mappings such that F has the mixed gmonotone property, and let there exist ${x}_{0},{y}_{0}\in X$ such that $g{x}_{0}\u2aafF({x}_{0},{y}_{0})$ and $F({y}_{0},{x}_{0})\u2aafg{y}_{0}$. Suppose that there exists $k\in [0,\frac{1}{2})$ such that for all $x,y,u,v,w,z\in X$ the following holds:
for all $gw\u2aafgu\u2aafgx$ and $gy\u2aafgv\u2aafgz$, where either $gu\ne gz$ or $gv\ne gw$. Assume the following hypotheses:

(i)
$F(X\times X)\subseteq g(X)$,

(ii)
$g(X)$ is Gcomplete,

(iii)
g is Gcontinuous and commutes with F.
Then F and g have a coupled coincidence point, that is, there exists $(x,y)\in X\times X$ such that $gx=F(x,y)$ and $gy=F(y,x)$. If $gu=gz$ and $gv=gw$, then F and g have a common fixed point, that is, there exists $x\in X$ such that $gx=F(x,x)=x$.
Theorem 16 If in the above theorem, we replace the condition (ii) by the assumption that X is gordered complete, then we have the conclusions of Theorem 15.
We next give the definition of Gcompatible mappings inspired by the definition of compatible mappings in [13].
Definition 17 Let $(X,G)$ be a Gmetric space. The mappings $F:X\times X\to X$, $g:X\to X$ are said to be Gcompatible if
and
where $\{{x}_{n}\}$ and $\{{y}_{n}\}$ are sequences in X such that ${lim}_{n\to \mathrm{\infty}}F({x}_{n},{y}_{n})={lim}_{n\to \mathrm{\infty}}g{x}_{n}=x$ and ${lim}_{n\to \mathrm{\infty}}F({y}_{n},{x}_{n})={lim}_{n\to \mathrm{\infty}}g{y}_{n}=y$ for all $x,y\in X$ are satisfied.
In this paper, we aim to extend the results on coupled fixed points mentioned above. Our results improve, enrich and extend some existing theorems in the literature. We also give examples to illustrate our results. This paper can also be considered as a continuation of the works of Berinde [11, 12].
2 Main results
We start with an example which shows the weakness of Theorem 15.
Example 18 Let $X=\mathbb{R}$. Define $G:X\times X\times X\to [0,\mathrm{\infty})$ by
for all $x,y,z\in X$. Let ⪯ be usual order. Then $(X,G)$ is a Gmetric space. Define a map $F:X\times X\to X$ by $F(x,y)=\frac{1}{8}x+\frac{5}{8}y$ and $g:X\to X$ by $g(x)=\frac{7x}{8}$ for all $x,y\in X$. Let $x=u=z$. Then we have
and
It is clear that there is no $k\in [0,\frac{1}{2})$ for which the statement (1.4) of Theorem 15 holds. Notice, however, that $(0,0)$ is the unique coupled coincidence point of F and g. In fact, it is a common fixed point of F and g, that is, $F(0,0)=g0=0$.
We now state our first result which successively guarantees the existence of a coupled coincidence point.
Theorem 19 Let $(X,\u2aaf)$ be a partially ordered set and $(X,G)$ be a Gcomplete Gmetric space. Let $F:X\times X\to X$ and $g:X\to X$ be two mappings such that F has the mixed gmonotone property on X and
for all $x,y,u,v,z,w\in X$ with $gx\u2ab0gu\u2ab0gw$, $gy\u2aafgv\u2aafgz$. Assume that $F(X\times X)\subset g(X)$, g is Gcontinuous and that F and g are Gcompatible mappings. Suppose further that either

(a)
F is continuous or

(b)
$(X,G,\u2aaf)$ is gordered complete.
Suppose also that there exist ${x}_{0},{y}_{0}\in X$ such that $g{x}_{0}\u2aafF({x}_{0},{y}_{0})$ and $F({y}_{0},{x}_{0})\u2aafg{y}_{0}$. If $k\in [0,1)$, then F and g have a coupled coincidence point, that is, there exists $(x,y)\in (X\times X)$ such that $g(x)=F(x,y)$ and $g(y)=F(y,x)$.
Proof Let ${x}_{0},{y}_{0}\in X$ be such that $g{x}_{0}\u2aafF({x}_{0},{y}_{0})$ and $F({y}_{0},{x}_{0})\u2aafg{y}_{0}$. Using the fact that $F(X\times X)\subset g(X)$, we can construct two sequences $\{{x}_{n}\}$ and $\{{y}_{n}\}$ in X in the following way:
We shall prove that for all $n\ge 0$,
Since $g{x}_{0}\u2aafF({x}_{0},{y}_{0})$ and $F({y}_{0},{x}_{0})\u2aafg{y}_{0}$ and $g{x}_{1}=F({x}_{0},{y}_{0})$ and $F({y}_{0},{x}_{0})=g{y}_{0}$, we have $g{x}_{0}\u2aafg{x}_{1}$ and $g{y}_{1}\u2aafg{y}_{0}$, that is, (2.5) holds for $n=0$. Assume that (2.5) holds for some $n>0$. Since F has the mixed gmonotone property, from (2.4), we have
and
By mathematical induction, it follows that (2.5) holds for all $n\ge 0$, that is,
and
If there exists ${n}_{0}\in \mathbb{N}$ such that $(g{x}_{{n}_{0}+1},g{y}_{{n}_{0}+1})=(g{x}_{{n}_{0}},g{y}_{{n}_{0}})$, then F and g have a coupled coincidence point. Indeed, in that case we would have
We suppose that $(g{x}_{n+1},g{y}_{n+1})\ne (g{x}_{n},g{y}_{n})$ for all $n\in \mathbb{N}$. More precisely, we assume that either $g{x}_{n+1}=F({x}_{n},{y}_{n})\ne g{x}_{n}$ or $g{y}_{n+1}=F({y}_{n},{x}_{n})\ne g{y}_{n}$.
For $n\in \mathbb{N}$, we set
Then by using (2.3) and (2.6), for each $n\in \mathbb{N}$, we have
which yields that
Now, for all $m,n\in \mathbb{N}$ with $m>n$, by using rectangle inequality (G5) of Gmetric and (2.10), we get
which yields that
Then by Proposition 6, we conclude that the sequences $\{g{x}_{n}\}$ and $\{g{y}_{n}\}$ are GCauchy.
Noting that $g(X)$ is Gcomplete, there exist $x,y\in g(X)$ such that $\{g{x}_{n}\}$ and $\{g{y}_{n}\}$ are Gconvergent to x and y respectively, i.e.,
Since F and g are Gcompatible mappings, by (2.11), we have
Suppose that the condition (a) holds. For all $n>0$, we have
Letting $n\to \mathrm{\infty}$ in the above inequality, using (2.11), (2.12) and the continuities of F and g, we have
Hence, we derive that $gx=F(x,y)$ and $gy=F(y,x)$, that is, $(x,y)\in {X}^{2}$ is a coupled coincidence point of F and g. Suppose that the condition (b) holds. By (2.8), (2.9) and (2.11), we have
Due to the fact that F and g are Gcompatible mappings and g is continuous, by (2.11) and (2.12), we have
Keeping (2.15) and (2.16) in mind, we consider now
Letting $n\to \mathrm{\infty}$ in the above inequality, by using (2.15), (2.16) and the continuity of g, we conclude that
By (G1), we have $gx=F(x,y)$ and $gy=F(y,x)$. Consequently, the element $(x,y)\in X\times X$ is a coupled coincidence point of the mappings F and g. □
Corollary 20 Let $(X,\u2aaf)$ be a partially ordered set and $(X,G)$ be a Gmetric space such that $(X,G)$ is Gcomplete. Let $F:X\times X\to X$ and $g:X\to X$ be two mappings such that F has the mixed gmonotone property on X and
for all $x,y,u,v\in X$ with $gx\u2ab0gu$, $gy\u2aafgv$. Assume that $F(X\times X)\subset g(X)$, the selfmapping g is Gcontinuous and F and g are Gcompatible mappings. Suppose that either

(a)
F is continuous or

(b)
$(X,G,\u2aaf)$ is gordered complete.
Suppose also that there exist ${x}_{0},{y}_{0}\in X$ such that $g{x}_{0}\u2aafF({x}_{0},{y}_{0})$ and $g{y}_{0}\u2ab0F({y}_{0},{x}_{0})$. If $k\in [0,1)$, then F and g have a coupled coincidence point.
Proof It is sufficient to take $z=u$ and $w=v$ in Theorem 19. □
Corollary 21 Let $(X,\u2aaf)$ be a partially ordered set and $(X,G)$ be a Gmetric space such that $(X,G)$ is Gcomplete. Let $F:X\times X\to X$ and $g:X\to X$ be two mappings such that F has the mixed gmonotone property on X and
for all $x,y,u,v\in X$ with $gx\u2ab0gu\u2ab0gw$, $gy\u2aafgv\u2aafgz$. Assume that $F(X\times X)\subset g(X)$ and that the selfmapping g is Gcontinuous and commutes with F. Suppose that either

(a)
F is continuous or

(b)
$(X,G,\u2aaf)$ is gordered complete.
Suppose further that there exist ${x}_{0},{y}_{0}\in X$ such that $g{x}_{0}\u2aafF({x}_{0},{y}_{0})$ and $g{y}_{0}\u2ab0F({y}_{0},{x}_{0})$. If $k\in [0,1)$, then F and g have a coupled coincidence point.
Proof Since g commutes with F, then F and g are Gcompatible mappings. Thus, the result follows from Theorem 19. □
Corollary 22 Let $(X,\u2aaf)$ be a partially ordered set and $(X,G)$ be a Gmetric space such that $(X,G)$ is Gcomplete. Let $F:X\times X\to X$ and $g:X\to X$ be two mappings such that F has the mixed gmonotone property on X and
for all $x,y,u,v\in X$ with $gx\u2ab0gu$, $gy\u2aafgv$. Assume that $F(X\times X)\subset g(X)$ and that g is Gcontinuous and commutes with F. Suppose that either

(a)
F is continuous or

(b)
$(X,G,\u2aaf)$ is gordered complete.
Assume also that there exist ${x}_{0},{y}_{0}\in X$ such that $g{x}_{0}\u2aafF({x}_{0},{y}_{0})$ and $g{y}_{0}\u2ab0F({y}_{0},{x}_{0})$. If $k\in [0,1)$, then F and g have a coupled coincidence point.
Proof Since g commutes with F, then F and g are Gcompatible mappings. Thus, the result follows from Corollary 20. □
Letting $g=I$ in Theorem 19 and in Corollary 20, we get the following results.
Corollary 23 Let $(X,\u2aaf)$ be a partially ordered set and $(X,G)$ be a Gmetric space such that $(X,G)$ is Gcomplete. Let $F:X\times X\to X$ be a mapping having the mixed monotone property on X and
for all $x,y,u,v,z,w\in X$ with $x\u2ab0u\u2ab0w$, $y\u2aafv\u2aafz$. Suppose that either

(a)
F is continuous or

(b)
$(X,G,\u2aaf)$ is ordered complete.
Suppose also that there exist ${x}_{0},{y}_{0}\in X$ such that ${x}_{0}\u2aafF({x}_{0},{y}_{0})$ and ${y}_{0}\u2ab0F({y}_{0},{x}_{0})$. If $k\in [0,1)$, then F has a coupled fixed point.
Corollary 24 Let $(X,\u2aaf)$ be a partially ordered set and $(X,G)$ be a Gmetric space such that $(X,G)$ is Gcomplete. Let $F:X\times X\to X$ be a mapping having the mixed monotone property on X and
for all $x,y,u,v\in X$ with $x\u2ab0u$, $y\u2aafv$. Suppose that either

(a)
F is continuous or

(b)
$(X,G,\u2aaf)$ is ordered complete.
Suppose further that there exist ${x}_{0},{y}_{0}\in X$ such that ${x}_{0}\u2aafF({x}_{0},{y}_{0})$ and ${y}_{0}\u2ab0F({y}_{0},{x}_{0})$. If $k\in [0,1)$, then F has a coupled fixed point.
Example 25 Let us recall Example 18. We have
and
It is clear that there any $k\in [\frac{6}{7},1)$ provides the statement (2.3) of Theorem 19.
Notice that $(0,0)$ is the unique coupled coincidence point of F and g which is also common coupled fixed point, that is, $F(0,0)=g0=0$.
Example 26 Let $X=\mathbb{R}$. Define $G:X\times X\times X\to [0,\mathrm{\infty})$ by
for all $x,y,z\in X$. Let ⪯ be usual order. Then $(X,G)$ is a Gmetric space.
Define a map $F:X\times X\to X$ by
and $g:X\to X$ by $g(x)={x}^{3}$ for all $x,y\in X$. Then $F(X\times X)=X=g(X)$. We observe that
and
then the statement (2.3) of Theorem 19 is satisfied for any $k\in (\frac{3}{4},1)$ and $(0,0)$.
Notice that if we replace the condition (2.3) of Theorem 19 with the condition (1.4) of Theorem 15 [21], that is,
where $k\in [0,\frac{1}{2})$, then the coupled coincidence point exists even though the contractive condition is not satisfied.
More precisely, consider $x=u=z$. Then we have
and
It is clear that the condition (2.27) holds for $k>\frac{5}{8}$.
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Acknowledgements
The second author gratefully acknowledges the support provided by the Department of Mathematics and Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT) during his stay at the Department of Mathematical and Statistical Sciences, University of Alberta as a visitor for the short term research.
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Keywords
 coupled fixed point
 coupled coincidence point
 mixed gmonotone property
 ordered set
 Gmetric space
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